Theory of Random Sets

In probability theory, we can not go very far and not mention the name of Kolmogorov. Together with the foundations of the modern probability theory Kolmogorov introduced the concept of a random set. The preface provides a small account on the history of random sets together with relevant references for further reading. The attractiveness of random sets, for a general reader at least, is in the applications. Naturally, we can think of numerous applications of random sets where some are spatial statistics, image analysis, mathematical economics, electrical engineering, convex and integral geometry, and others.

The book is written in a theorem-proof style, where the proofs are quite detailed and clearly presented. The author really does spend a lot of time to produce legible proofs that are easy to follow. Furthermore, to enable easier reading, the author provides visual illustrations where necessary. These should be of great help to graduate students pursuing research in this field. Each section (chapter) of the book ends with notes. The provide additional references for each subsection and small theoretical sidebars and  explanations. One thing that is really valuable for a beginner researcher or a graduate student are numerous notes on small but important issues such as who first suggested the concept of a Gaussian random set or who studied the convergence of random broken lines, etc.

This is a highly mathematical text. The prospective reader should be firmly trained for a book in the theorem-proof style that assumes background knowledge in modern probability, stochastic processes, analysis, etc. Several appendices at the end of the book revuew the necessary background in topological and linear spaces, measure theory, probability theory, capacities, convex sets, etc. As this is more of a reference text there are no "standard" problems, but rather the author provides "open" problems to enhance the reader's attractiveness to the subject.

The book is divided into five main chapters: Random Closed Sets and Capacity Functionals, Expectations of Random Sets, Minkowski Addition, Unions of Random Sets, and Random Sets and Random Functions. Each of the five chapters represents important concepts in the theory of random sets. The capacity functional is explained in first chapter. It provides the definition of a random closed set. Then the Choquet theorem is stated and proved in detail. The concept of the capacity functional is given as well as the analytic theory, convergence concepts, some applications to point processes, random measures, etc.

Chapter 2 covers expectation for random closed sets. The selection expectation is given in the first part of the chapter. Expectation on lattices and in metric spaces is also briefly covered at the end of the chapter. Chapter 3 covers Minkowski Addition, which closely relates to section 2. The strong law of large numbers and the Central Limit theorem are explained in detail. The union of random sets, which is closely related to extremes of random variables, is covered in chapter 4.  Chapter 5 describes the connection between stochastic processes and random sets.

Reading the text one can really get a feeling for the author's extensive knowledge of the subject. "Theory of Random Sets" is an excellent reference for a serious researcher. Apart from the theoretical explanations the author provides a vast bibliography on the subject, which is completely searchable on author's website. The book should be read and even more studied by any researcher or a student considering  research in this field.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.